Morse theory for periodic solutions of hamiltonian systems and the maslov index

Abstract

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over q ( M ) . The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every I-periodic solution has at least one Floquet multiplier which is not equal to 1.

Periodic orbits of twisted geodesic flows and the Weinstein–Moser theorem

“…However, looking at the question from the perspective of the Conley conjecture (see [FrHa,Gi9,Hi,SZ]) rather than of the Arnold conjecture, one can expect every low level of K to carry infinitely many periodic orbits (not necessarily short), provided that dim M ≥ 2 and M is symplectically aspherical. An indication that this may indeed be the case is given by Proposition 1.5.…”

“…The standard argument shows that ∂ 2 ǫ = 0 whenever ǫ > 0 and F −F C 2 are small enough, and, moreover, the resulting local Floer homology spaces HF * (F, Σ) are independent of ǫ > 0,F and J; see [Fl5] and also, e.g., [BPS,Gi9,HS,McSa,Poz,Sa,SZ]. (To be more precise, here we need to require that ǫ < ǫ 0 (U, J) and F −F C 2 < δ 0 (U, J, ǫ).…”

Positive topological entropy of Reeb flows on spherizations

“…By counting these solutions one can therefore define the Floer chain map [15,40,16,7]. The induced continuation homomorphism…”

Symplectic rigidity, symplectic fixed points, and global perturbations of Hamiltonian systems

“…With this assumption and utilizing our assumption that c 1 (T M )| π2(M) = 0, the elements of P(H) are graded by their Conley-Zehnder index, µ CZ ; see [22]. The key result concerning moduli spaces M(x − , x + , J, H) is the following, see [21], Theorem 3.…”

“…where m(x) is the Morse index of x, we refer to [22], for this and other facts concerning the properties of the Conley-Zehnder index. That is to say that for sufficiently small ε > 0 and large λ, and x 0 -a critical point of H λ with Morse index l, then…”